Igor Aleiner (Columbia) Theory of Quantum Dots as Zero-dimensional Metallic Systems Physics of the Microworld Conference, Oct. 16 (2004) Collaborators: B.L.Altshuler (Princeton) P.W.Brouwer (Cornell) V.I.Falko (Lancaster, UK) L.I. Glazman (Minnesota) I.L. Kurland (Princeton)

Outline: Quantum dot (QD) as zero dimensional metal Random Matrix theory for transport in quantum dots a)Non-interacting “standard models”. b)Peculiar spin-orbit effects in QD based on 2D electron gas. Interaction effects: a)Universal interaction Hamiltonian; b)Mesoscopic Stoner instability; c)Coulomb blockade (strong, weak, mesoscopic); d)Kondo effect.

“Quantum dot” used in two different contents: “Artificial atom” Description requires exact diagonalization. “Artificial nucleus” Statistical description is allowed !!! (Kouwnehoven group (Delft)) Number of electrons: 1) (Marcus group (Harvard)) 2) For the rest of the talk:

Random Matrix Theory for Transport in Quantum Dots 2DEG QD L Energy scales 2DEG Level spacing Thouless Energy Conductance Assume:

Statistics of transport is determined only by fundamental symmetries !!! Reviews: Beenakker, Rev. Mod. Phys. 69, 731 (1997) Alhassid, Rev. Mod. Phys. 72, 895 (2000) Aleiner, Brouwer, Glazman, Phys. Rep., 309 (2002) Original Hamiltonian: Confinement, disorder, etc RMT

No magnetic field, no SO Magnetic field, no SO No magnetic field, strong SO Magnetic field + SO

I V Conductance of chaotic dot classical Mesoscopic fluctuations Weak localization Jalabert, Pichard, Beenakker (1994) Baranger, Mello (1994)

I V Conductance of chaotic dot classical Mesoscopic fluctuations Weak localization [Altshuler, Shklovskii (1986)] Universal quantum corrections

Peculiar effect of the spin-orbit interaction Naively: SO But the spin-orbit interaction in 2D is not generic.

Spin-orbit interaction in GaAs/AlGaAs (001) 2DEG - spin-orbit lengths [001] Rashba term Dresselhaus term Dyakonov-Perel spin relaxation

Approximate symmetries of SO in QD Aleiner, Fal’ko (2001) T - invariance But Spin dependent flux Spin relaxation rate Mathur, Stone (1992) Khaetskii, Nazarov (2000) Meir, Gefen, Entin-Wohlman (1989) Lyanda-Geller, Mirlin (1994)

Energy scales: Brouwer, Cremers,Halperin (2002) May be violated for

Effect of Zeeman splitting Orthogonal, !!! But no spin degeneracy; spins mixed: New energy scale:

6 possible symmetry classes:

Orbital effect of the magnetic field

Observed in Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian Energies smaller than Thouless energy: Random matrix ???? In nuclear physics: from shell model random

Universal Interaction Hamiltonian Energies smaller than Thouless energy: are NOT random !!! Kurland, Aleiner, Altshuler (2000) Only invariants compatible with the circular symmetry

Universal Interaction Hamiltonian Energies smaller than Thouless energy: Valid if: 1) 2) Fundamental symmetries are NOT broken at larger energies Random matrix Not random

Universal Interaction Hamiltonian Energies smaller than Thouless energy: Valid if: 1) 2) Fundamental symmetries are NOT broken at larger energies One-particle levels determined by Wigner – Dyson statistics Interaction with additional conservations Zero dimensional Fermi liquid

Universal Interaction Hamiltonian Analogy with soft modes in metals Singlet electron-hole channel. Triplet electron-hole channel. Particle-particle (Cooper) channel.

Universal Interaction Hamiltonian Cooper Channel: Renormalization: Normal Superconducting (e.g. Al grains)

Universal Interaction Hamiltonian Triplet Channel: is NOT renormalized But may lead to the spin of The ground state S > ½.

Mesoscopic Stoner Instability Kurland, Aleiner, Altshuler (2000) Also Brouwer, Oreg, Halperin (2000) vs. Energy of the ground state: NO randomness NO interactions FM instability Stoner (1935) random with known from RMT correlation functions Spin is finite even for Typical S: Does not scale with the size of the system

Universal Interaction Hamiltonian Singlet Channel: is NOT renormalized gate voltage But Q: What is charge degeneracy of the ground state

- half-integer Otherwise degeneracy gap (isolated dot)

Coulomb blockade of electron transport Term introduced by Averin and Likharev (1986); Effect first discussed by C.J. Gorter (1951). For tunneling contacts: Charge degeneracy Charge gap

Small quantum dots (~ 500 nm) M. Kastner, Physics Today (1993) E.B. Foxman et al., PRB (1993) conductance (e 2 /h) gate voltage (mV) In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II) Strong CB Weak CB Mesoscopic CB (reflectionless contacts) Random phase but not period.

Courtesy of C.Marcus Statistical description of strong CB: Theory: Peaks: Jalabert, Stone, Alhassid (1992); Valleys: Aleiner, Glazman (1996); Reasonable agreement, But problems with values of the correlation fields

Mesoscopic Coulomb Blockade Based on technique suggested by: Matveev (1995); Furusaki, Matveev (1995); Flensberg (1993). Aleiner, Glazman (1998)

Experiment: Cronenwett et. al. (1998) Suppression By a factor of 5.3 Th: Predicted 4.

Even-Odd effect due to Kondo effect Spin degeneracy in odd valleys: Effective Hamiltonian: magnetic impurity local spin density of conduction electrons Predicted: Glazman, Raikh (1988) Ng, Lee (1988)

Observation: D. Goldhaber-Gordon et al. (MIT-Weizmann) S.M. Cronenwett et al. (TU Delft) J. Schmid et al. Stuttgart) 1998 van der Wiel et al. (2000) 200 nm 15 mK800 mK

Conclusions 1)Random matrix is an adequate description for the transport in quantum dots if underlying additional symmetries are properly identified. 2) Interaction effects are described by the Universal Hamiltonian (“0D Fermi Liquid”)